In this dissertation, a theory of topological K-theory of dg-categories relative to an arbitrary base scheme is developed. This is then used to study the topological K-theory of Azumaya algebras and relate it to the topological notion of twisted K-theory.
Advisors/Committee Members: Shipley, Brooke (advisor), Antieau, Benjamin (advisor), Gillet, Henri (committee member), Gepner, David (committee member), Heller, Jeremiah (committee member), Shipley, Brooke (chair).